Infinitely many independent functions that are only frequency localized?

Question

A function $f \in L^2(\mathbb R^d)$ will be called $K$-frequency localized if the following inequality holds $$\int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 x^2 \ dx \le K \int_{\mathbb R^d} \lvert \widehat{f}(x) \rvert^2 \ dx.$$

On the other hand, we say a function $f \in L^2(\mathbb R^d)$ is not spatially localized if

$$\int_{\mathbb R^d} \lvert f(x) \rvert^2 x^2 \ dx =\infty.$$

I am wondering: Are there infinitely-many $L^2$ linearly-independent functions that are $K$-frequency localized for some (any) fixed $K >0$ that are not spatially localized?

I feel the answer is yes and it should follow from general Banach space arguments, but I do not really see how to argue this.

Answer 1

For all $a\in(0,K)$, the function $\hat 1_{[0,a]}$ is $K$-frequency localized, yet the function itself decays like $1/x$ as $x\to\infty$ (as easily seen by integration by parts), so it is not spatially localized.